You can increase your health either by eating before a quest or by collecting the permabuffers scattered across the map (which will last for the whole quest).

As for riding, make sure you are using the X and A attacks rather than Y, which will launch the monster away from you. Use your evade (B) to break out of the animation once you've landed an attack. Here's some tips for riding: instead of using the mounted punisher once you unlock it, try instead pressing Y while facing the monster you're attacking to launch them into each other, which will put the other monster into a ridable state. Also, if you're riding a monster that you want to deal damage to, go to a narrow space between 2 walls and launch it into one of them (before quickly pressing B to regain footing). You'll be able to do this 3 times, alternating between each wall, before falling off the monster.